Vacancy properties in Cu 3 Au-type ordered fcc alloys
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A theory of vacancy formation in Cu3Au-type ordered fee alloys is presented. The present theory, which is based on the pairwise bonding model, is found to be in good agreement with the experimentally observed vacancy properties in ordered CU3A11 and Ni3Al. Various shortcomings in the previous theoretical calculations have also been identified.
I. INTRODUCTION
Vacancy properties in metals and alloys are of interest since atomic diffusion is usually controlled by vacancies. There have been several previous theoretical calculations on the vacancy concentrations in Cu3Autype ordered fee alloys.1^ In most of these calculations, however, assumptions have been made that can be shown to be either unreasonable or very restrictive in applicability. In this paper, we present a theory that is free from these shortcomings. Like most of the previous calculations, the present theory is also based on the pairwise bonding model. In recent years we have shown that theories based on the pairwise bonding model can describe satisfactorily many of the observed defect and diffusion properties in certain ordered alloys as well as in disordered alloys.5"9 II. THEORY
We assume that the total energy of an alloy is given by the sum of the nearest-neighbor bond energies. When a vacancy is created by removing an atom from inside a crystal and placing it on the surface, the electrons and ions surrounding the vacancy relax in order to lower the crystal free energy. The lowering of the free energy due to the relaxation around a vacancy can be very large. In the present theory we take into account this relaxation effect approximately in the free energy expression. We denote the relaxation energy of an A(B) atom surrounding a vacancy as eAV(eBV); i.e., we assume that a fictitious attractive bonding is established between an A(B) atom and a vacancy. The derivation of the theory is then similar to those for the other ordered alloys.8'9 The A3B alloy is assumed to consist of the sites a and f3 of two sublattices. Any particular site may be occupied by an A atom, a B atom, or by a vacancy V. We define 3(N + n) = number of a sites, (N + n) = number of (3 sites, 3(1 - 6)N = number of A atoms, (1 + 36)N = number of B atoms, An = number of vacancies, and NAa = number of A species on a sites. We then have NAa = 3(1 - S)N - N;'A/3
NBf3 = 1 + 3S)N - NBa 3SN + NAp - NBa
(lc)
NVI3 = n - 36N - NAp + NBa.
(Id)
NVa = 3n
In these equations n/(N + n) is the vacancy concentration per site, and 38/2 is the excess of B over A atoms relative to ideal stoichiometry, as a fraction of the total number of atoms, AN. The configurational part of the free energy at a temperature T can be written as F = E-TS
atoms, 8NBa/3{N
+ n)B atoms, and 8NVa/3(N
+ n)
vacancies on the a sublattice, and by ANA0/(N + n) A atoms, ANB0/(N + n) B atoms, and ANv/3/(N + n) vacancies on the (3 sublattice. Thus the total energy will be given by E = -
(la)
Downloaded: 11 Mar 2015
(2)
where E is the total bond energy in the crystal and S is the configurational entropy. If we denote the
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